Symmetry of ground states

for a semilinear elliptic system

Author:
Henghui Zou

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1217-1245

MSC (1991):
Primary 35B40, 35J60

DOI:
https://doi.org/10.1090/S0002-9947-99-02526-X

Published electronically:
September 20, 1999

MathSciNet review:
1675167

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let and consider the following system

By using the Alexandrov-Serrin moving plane method, we show that under suitable assumptions every *slow decay* solution of (I) must be radially symmetric.

**1.**A.D. Alexandrov,*A characteristic property of the spheres*, Ann. Mat. Pura Appl.**58**(1962), 303-315. MR**26:722****2.**M.-F. Bidaut-Veron and L. Veron,*Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations*, Invent. Math.,**106**(1991), 489-539; Erratum, Invent. Math.**112**(1993). MR**93a:35045**; MR**94b:53069****3.**L. Caffarelli, B. Gidas and J. Spruck,*Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth*, Comm. Pure Appl. Math.,**42**(1989), 271-297. MR**90c:35075****4.**R. Courant and D. Hilbert,*Methods of Mathematical Physics*, Vols. I and II, Interscience-Wiley, New York, 1962. MR**16:426a**; MR**25:4216****5.**D.G. de Figueiredo and P.L. Felmer,*A Liouville-type theorem for elliptic systems*, Ann. Scuola Norm. Sup. Pisa**21**(1994), no.3, 387-397. MR**95m:35009****6.**M. Escobedo and M. Herrero,*Boundedness and blow-up for a semilinear reaction-diffusion system*, J. Diff. Equation**89**(1991), 176-202. MR**91j:35040****7.**M. Escobedo and H. Levine,*Critical blow-up and global existence numbers for a weakly coupled system of reaction-diffusion equations*, Arch. Rational Mech. Anal.**129**(1995), 47-100. MR**96d:35063****8.**B. Gidas, W. Ni and L. Nirenberg,*Symmetry of positive solutions of nonlinear ellipttic equations in*, Math. Anal. Appl., Part A, Academic Press, 1981, pp. 369-402. MR**84a:35083****9.**D. Gilbarg and N. Trudinger,*Elliptic Partial Differential Equations of Second Order*, Springer-Verlag, New York, 1983 (second edition), pp. 369-402. MR**86c:35035****10.**H.A. Levine,*The role of critical exponents in blow-up theorems*, SIAM Review,**3**(1990), 262-298. MR**91j:35135****11.**E. Mitidieri,*Non-existence of positive solutions of semilinear elliptic systems in*, Quaderno Matematico,**285**(1992).**12.**J. Serrin,*A symmetry problem in potential theory,*Arch. Rational Mech. Anal.**43**(1971), pp.304-318. MR**48:11545****13.**J. Serrin and H. Zou,*Non-existence of positive solutions of semilinear elliptic systems*, Discourses in Math. Appl., 3, Texas A&M Univ., College Station, 1994, pp. 55-68. MR**97k:35065****14.**J. Serrin and H. Zou,*Non-existence of positive solutions of Lane-Emden systems*, Differential and Integral Equations,**9**(1996), no.4, 635-653. MR**97f:35056****15.**J. Serrin and H. Zou,*Existence of positive solutions of the Lane-Emden system*, Atti del Sem. Mat. Fis. Univ. Modena**43**(1998), Suppl., 369-380. CMP**99:01****16.**L. Simon,*Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems*, Ann. Math.**118**(1983), 525-571. MR**85b:58121****17.**L. Simon,*Isolated singularities of extrema of geometric variational problems*, Lecture Notes in Math.**1161**(1985), 206-277. MR**87d:58045****18.**W. Troy,*Symmetry properties in systems of semilinear elliptic equations*, J. Differential Equations,**42**(1981), 400-413. MR**83b:35051****19.**H. Zou,*Symmetry of positive solutions of in ,*J. Differential Equations**120**(1995), no.1, 46-88. MR**96h:35055****20.**H. Zou,*Slow decay and the Harnack inequality for positive solutions of in ,*Differential and Integral Equations**8**(1995), no.6, 1355-1368. MR**96b:35057****21.**H. Zou,*Symmetry of ground states of semilinear equations with mixed Sobolev growth*, Indiana Univ. Math. J.**45**(1996), no.1, 221-240. MR**97h:35051****22.**H. Zou, in preparation.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
35B40,
35J60

Retrieve articles in all journals with MSC (1991): 35B40, 35J60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-99-02526-X

Received by editor(s):
April 4, 1997

Received by editor(s) in revised form:
October 20, 1997

Published electronically:
September 20, 1999

Additional Notes:
Research supported in part by NSF Grants DMS-9418779 and DMS-9622937, an Alabama EPSCoR grant and a faculty research grant of the University of Alabama at Birmingham

Article copyright:
© Copyright 1999
American Mathematical Society